Calculus Examples

$9 \ln (8 \ln (x))$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 \ln (8 \ln (x))$ with respect to $x$ is $9 \frac{d}{d x} [\ln (8 \ln (x))]$ .

$9 \frac{d}{d x} [\ln (8 \ln (x))]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 8 \ln (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $8 \ln (x)$ .

$9 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [8 \ln (x)])$

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$9 (\frac{1}{u} \frac{d}{d x} [8 \ln (x)])$

Step 2.3

Replace all occurrences of $u$ with $8 \ln (x)$ .

$9 (\frac{1}{8 \ln (x)} \frac{d}{d x} [8 \ln (x)])$

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Combine $\frac{1}{8 \ln (x)}$ and $9$ .

$\frac{9}{8 \ln (x)} \frac{d}{d x} [8 \ln (x)]$

Step 3.2

Since $8$ is constant with respect to $x$ , the derivative of $8 \ln (x)$ with respect to $x$ is $8 \frac{d}{d x} [\ln (x)]$ .

$\frac{9}{8 \ln (x)} (8 \frac{d}{d x} [\ln (x)])$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $8$ and $\frac{9}{8 \ln (x)}$ .

$\frac{8 \cdot 9}{8 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.2

Multiply $8$ by $9$ .

$\frac{72}{8 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.3

Cancel the common factor of $72$ and $8$ .

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Step 3.3.3.1

Factor $8$ out of $72$ .

$\frac{8 \cdot 9}{8 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $8$ out of $8 \ln (x)$ .

$\frac{8 \cdot 9}{8 (\ln (x))} \frac{d}{d x} [\ln (x)]$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{8 \cdot 9}{8 \ln (x)} \frac{d}{d x} [\ln (x)]$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{9}{\ln (x)} \frac{d}{d x} [\ln (x)]$

Step 4

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{9}{\ln (x)} \cdot \frac{1}{x}$

Step 5

Combine fractions.

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Step 5.1

Multiply $\frac{9}{\ln (x)}$ by $\frac{1}{x}$ .

$\frac{9}{\ln (x) x}$

Step 5.2

Reorder terms.

$\frac{9}{x \ln (x)}$